# American Institute of Mathematical Sciences

April  2019, 13(2): 263-284. doi: 10.3934/ipi.2019014

## On finding the surface admittance of an obstacle via the time domain enclosure method

 Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashihiroshima 739-8527, Japan

Received  June 2017 Revised  August 2018 Published  January 2019

Fund Project: The author was partially supported by Grant-in-Aid for Scientific Research (C)(No 17K05331) of Japan Society for the Promotion of Science.

An inverse obstacle scattering problem for the electromagnetic wave governed by the Maxwell system over a finite time interval is considered. It is assumed that the wave satisfies the Leontovich boundary condition on the surface of an unknown obstacle. The condition is described by using an unknown positive function on the surface of the obstacle which is called the surface admittance. The wave is generated at the initial time by a volumetric current source supported on a very small ball placed outside the obstacle and only the electric component of the wave is observed on the same ball over a finite time interval. It is shown that from the observed data one can extract information about the value of the surface admittance and the curvatures at the points on the surface nearest to the center of the ball. This shows that a single shot contains a meaningful information about the quantitative state of the surface of the obstacle.

Citation: Masaru Ikehata. On finding the surface admittance of an obstacle via the time domain enclosure method. Inverse Problems and Imaging, 2019, 13 (2) : 263-284. doi: 10.3934/ipi.2019014
##### References:
 [1] N. G. Alexopoulos and G. A. Tadler, Accuracy of the Leontovich boundary condition for continuous and discontinuous surface impedances, J. Appl. Phys., 46 (2008), 3326-3332.  doi: 10.1063/1.322058. [2] C. A. Balanis, Antenna Theory, Analysis and Design, 3$^{rd}$ edition, Wiley-Interscience, Hoboken, New Jersey, 2005. [3] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover, New York, 1986. [4] M. Cheney and R. Borden, Fundamentals of Radar Imaging, CBMS-NSF, Regional Conference Series in Applied Mathematics, 79, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719291. [5] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. [6] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Spectral Theory and Applications, Vol. 3, Springer-Verlag, Berlin, 1990. [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edn, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [8] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308. [9] M. Ikehata, Extracting the geometry of an obstacle and a zeroth-order coefficient of a boundary condition via the enclosure method using a single reflected wave over a finite time interval, Inverse Problems, 30 (2014), 045011 (24pp). doi: 10.1088/0266-5611/30/4/045011. [10] M. Ikehata, New development of the enclosure method for inverse obstacle scattering, Chapter 6 in Inverse Problems and Computational Mechanics (eds. L. Marin, L. Munteanu, V. Chiroiu), Vol. 2,123–147, Editura Academiei, Bucharest, Romania, 2016. [11] M. Ikehata, The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain, Inverse Problems and Imaging, 10 (2016), 131-163.  doi: 10.3934/ipi.2016.10.131. [12] M. Ikehata, On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method, Inverse Problems and Imaging, 11 (2017), 99-123.  doi: 10.3934/ipi.2017006. [13] M. Ikehata, A remark on finding the coefficient of the dissipative boundary condition via the enclosure method in the time domain, Math. Meth. Appl. Sci., 40 (2017), 915-927.  doi: 10.1002/mma.4021. [14] B. V. Kapitonov, On exponential decay as $t\longrightarrow\infty$ of solutions of an exterior boundary value problem for the Maxwell system, Math. USSR Sbornik, 66 (1990), 475-498.  doi: 10.1070/SM1990v066n02ABEH001318. [15] A. Kirsch and F. Hettlich, The Mathematical Theory of Time-harmononic Maxwell's Equations, Expansion-, Integral-, and Variational Methods, Springer, 2015. doi: 10.1007/978-3-319-11086-8. [16] S. G. Krein and I. M. Kulikov, The Maxwell-Leontovich operator, Differentsial'nye Uravneniya, 5 (1969), 1275-1282. [17] P. D. Lax and R. S. Phillips, The scattering of sound waves by an obstacle, Comm. Pure and Appl. Math., 30 (1977), 195-233.  doi: 10.1002/cpa.3160300204. [18] J.-C. Nédélec, Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems, Springer, New York, 2001. doi: 10.1007/978-1-4757-4393-7. [19] B. O'Neill, Elementary Differential Geometry, Revised, 2nd Edition, Academic Press, Amsterdam, 2006. [20] K. Yosida, Functional Analysis, Third Edtition, Springer, New York, 1971.

show all references

##### References:
 [1] N. G. Alexopoulos and G. A. Tadler, Accuracy of the Leontovich boundary condition for continuous and discontinuous surface impedances, J. Appl. Phys., 46 (2008), 3326-3332.  doi: 10.1063/1.322058. [2] C. A. Balanis, Antenna Theory, Analysis and Design, 3$^{rd}$ edition, Wiley-Interscience, Hoboken, New Jersey, 2005. [3] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover, New York, 1986. [4] M. Cheney and R. Borden, Fundamentals of Radar Imaging, CBMS-NSF, Regional Conference Series in Applied Mathematics, 79, SIAM, Philadelphia, 2009. doi: 10.1137/1.9780898719291. [5] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. [6] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Sciences and Technology, Spectral Theory and Applications, Vol. 3, Springer-Verlag, Berlin, 1990. [7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edn, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [8] M. Ikehata, Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data, Inverse Problems, 15 (1999), 1231-1241.  doi: 10.1088/0266-5611/15/5/308. [9] M. Ikehata, Extracting the geometry of an obstacle and a zeroth-order coefficient of a boundary condition via the enclosure method using a single reflected wave over a finite time interval, Inverse Problems, 30 (2014), 045011 (24pp). doi: 10.1088/0266-5611/30/4/045011. [10] M. Ikehata, New development of the enclosure method for inverse obstacle scattering, Chapter 6 in Inverse Problems and Computational Mechanics (eds. L. Marin, L. Munteanu, V. Chiroiu), Vol. 2,123–147, Editura Academiei, Bucharest, Romania, 2016. [11] M. Ikehata, The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain, Inverse Problems and Imaging, 10 (2016), 131-163.  doi: 10.3934/ipi.2016.10.131. [12] M. Ikehata, On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method, Inverse Problems and Imaging, 11 (2017), 99-123.  doi: 10.3934/ipi.2017006. [13] M. Ikehata, A remark on finding the coefficient of the dissipative boundary condition via the enclosure method in the time domain, Math. Meth. Appl. Sci., 40 (2017), 915-927.  doi: 10.1002/mma.4021. [14] B. V. Kapitonov, On exponential decay as $t\longrightarrow\infty$ of solutions of an exterior boundary value problem for the Maxwell system, Math. USSR Sbornik, 66 (1990), 475-498.  doi: 10.1070/SM1990v066n02ABEH001318. [15] A. Kirsch and F. Hettlich, The Mathematical Theory of Time-harmononic Maxwell's Equations, Expansion-, Integral-, and Variational Methods, Springer, 2015. doi: 10.1007/978-3-319-11086-8. [16] S. G. Krein and I. M. Kulikov, The Maxwell-Leontovich operator, Differentsial'nye Uravneniya, 5 (1969), 1275-1282. [17] P. D. Lax and R. S. Phillips, The scattering of sound waves by an obstacle, Comm. Pure and Appl. Math., 30 (1977), 195-233.  doi: 10.1002/cpa.3160300204. [18] J.-C. Nédélec, Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems, Springer, New York, 2001. doi: 10.1007/978-1-4757-4393-7. [19] B. O'Neill, Elementary Differential Geometry, Revised, 2nd Edition, Academic Press, Amsterdam, 2006. [20] K. Yosida, Functional Analysis, Third Edtition, Springer, New York, 1971.
 [1] Masaru Ikehata. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. Inverse Problems and Imaging, 2017, 11 (1) : 99-123. doi: 10.3934/ipi.2017006 [2] Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems and Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131 [3] Lekbir Afraites. A new coupled complex boundary method (CCBM) for an inverse obstacle problem. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 23-40. doi: 10.3934/dcdss.2021069 [4] Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems and Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027 [5] Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems and Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211 [6] Annalena Albicker, Roland Griesmaier. Monotonicity in inverse scattering for Maxwell's equations. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022032 [7] Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems and Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019 [8] Masaru Ikehata, Mishio Kawashita. On finding a buried obstacle in a layered medium via the time domain enclosure method. Inverse Problems and Imaging, 2018, 12 (5) : 1173-1198. doi: 10.3934/ipi.2018049 [9] Peijun Li, Xiaokai Yuan. Inverse obstacle scattering for elastic waves in three dimensions. Inverse Problems and Imaging, 2019, 13 (3) : 545-573. doi: 10.3934/ipi.2019026 [10] Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems and Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77 [11] Lu Zhao, Heping Dong, Fuming Ma. Inverse obstacle scattering for acoustic waves in the time domain. Inverse Problems and Imaging, 2021, 15 (5) : 1269-1286. doi: 10.3934/ipi.2021037 [12] T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems and Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335 [13] M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473 [14] Ting Zhou. Reconstructing electromagnetic obstacles by the enclosure method. Inverse Problems and Imaging, 2010, 4 (3) : 547-569. doi: 10.3934/ipi.2010.4.547 [15] Mourad Sini, Nguyen Trung Thành. Inverse acoustic obstacle scattering problems using multifrequency measurements. Inverse Problems and Imaging, 2012, 6 (4) : 749-773. doi: 10.3934/ipi.2012.6.749 [16] Masaru Ikehata, Mishio Kawashita, Wakako Kawashita. On finding a buried obstacle in a layered medium via the time domain enclosure method in the case of possible total reflection phenomena. Inverse Problems and Imaging, 2019, 13 (5) : 959-981. doi: 10.3934/ipi.2019043 [17] Lu Zhao, Heping Dong, Fuming Ma. Time-domain analysis of forward obstacle scattering for elastic wave. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4111-4130. doi: 10.3934/dcdsb.2020276 [18] Laurent Bourgeois, Jérémi Dardé. A quasi-reversibility approach to solve the inverse obstacle problem. Inverse Problems and Imaging, 2010, 4 (3) : 351-377. doi: 10.3934/ipi.2010.4.351 [19] Laurent Bourgeois, Jérémi Dardé. The "exterior approach" to solve the inverse obstacle problem for the Stokes system. Inverse Problems and Imaging, 2014, 8 (1) : 23-51. doi: 10.3934/ipi.2014.8.23 [20] Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems and Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951

2021 Impact Factor: 1.483